Herman, et al,
The extension of a (binary) relation is a set of pairs.
The "domain" of a (binary) relation is (always) the set of all the first
pair elements in the extension
The "range" of a (binary) relation is the set of all the second pair
elements in the extension
But you are right, when I said "the domain of the relation for any given range is unique"
I meant "the domain of the relation for any given element of the range is
unique"
The real problem in expressing this for me was battling with the confusion
made in these documents between property and relation. A property is
distinctly, in mathematics and logic, NOT A RELATION! Yet we call it one.
This makes writing a definition really confusing.
Dr. Christopher A. Welty, Knowledge Structures Group
IBM T.J. Watson Research Center
PO Box 704, Yorktown Heights, NY 10598, USA
+1-914-784-7055 Fax: +1-914-784-6078
herman.ter.horst@philips.com
Sent by: www-webont-wg-request@w3.org
07/12/2002 04:33 AM
To: Christopher Welty/Watson/IBM@IBMUS, lynn.stein@olin.edu
cc: www-webont-wg@w3.org
Subject: Re: ISSUE:4.1 daml:uniqueProperty
In the mail of Chris of yesterday,
the words domain and range are used for sets and elements:
...
>An UnambiguousProp is a relation whose extension is restricted such that
>no object may appear more than once in the range, i.e. the domain of the
here the word range stands for a set and the word domain stands for an
element
>relation for any given range is unique.
here the word range stands for an element.
This is confusing. The basic practice in mathematics is to use the terms
domain and range only for sets.
This point is also linked to the choice of names:
>Proposed Resolution
>
>These are suggested alternate names for these features. To resolve the
>issue, we should vote on them. I will collect suggested alternate names
>until, say, the next telecon.
>
>UniqueProp: Functional, UniqueRange
>UnambiguousProp: InverseFunctional, UniqueDomain
The suggestions UniqueRange and UniqueDomain do not seem to be
consistent with the basic meaning given in mathematics to the words
domain and range.
In the standard definition, the domain and the range of a relation are
sets:
Given a relation, that is, a set of ordered pairs (x,y), the domain
of the relation is defined to be the set of all elements x that appear
as the first entry in at least one such ordered pair of the relation.
The range of the relation is defined to be the set of all elements y
that appear as second entry in at least one such ordered pair of the
relation.
This implies that for each relation, the domain is always uniquely
defined,
and also that the range is always uniquely defined:
The domain and range of a relation are uniquely defined sets associated to
the relation.
From this point of view, the terms UniqueRange and UniqueDomain do not
seem to add information.
The terms UniqueRangeElement and UniqueDomainElement would be more
consistent
with basic mathematical usage.
But these terms are longer, and, in my view, not the optimal choice:
they may suggest that the range and domain consist of exactly one element.
The terms SingletonRange and SingletonDomain, suggested yesterday by Lynn,
also wrongly suggest that the domain and range of the relation consist of
a singleton set, that is, of exactly one element
I vote in favor of the terms Functional and InverseFunctional.
Herman ter Horst